Math 503, Fall 2011
Topics Covered in Each Lecture
|
Lecture No |
Date |
Content |
Corresponding Reading material |
| 1 | Sep. 19 | Basic structure of mathematics and natural sciences, abstract versus applied mathematics, mathematical theories, basics of logic, sets and their elementary properties, relations | Pages 1-7 of "A First Course in Linear Algebra" |
| 2 | Sep. 21 | Equivalence relations and classification schemes. Functions: Image and inverse image of subsets under a function, domain and range, one-to-one and onto functions, bijections, composition of functions, invertible functions, equivalent sets | Pages 7-11 of "A First Course in Linear Algebra" |
|
3 |
Sep. 28 | Plane vectors, componentwise addition and multiplication, complex multiplication and complex number system, algebraic properties of complex numbers; modulus, argument, principal argument, and complex-conjugate of a complex number... | Pages 13-18 of "A First Course in Linear Algebra" & pages 602-608 of "Kreyszig" |
| 4 | Sep. 30 | Complex sequences and series and their convergence, The exponential of a complex number, Euler's formula and polar representation of complex numbers; Linear Algebra, plane vectors and their algebraic properties, generalization to R^n, Real and complex abstract vector spaces, notation and some basic properties, examples: Trivial vector spaces, Function spaces | Pages 18-21 of "A First Course in Linear Algebra" |
| 5 | Oct. 03 | Vector space of polynomials, Subspaces of a vector space, linear combination, span of a subset, characterization theorem for the span, Linear independence and the characterization theorem for linear independence | Pages 21-27 of "A First Course in Linear Algebra" |
| 6 | Oct. 04 | Examples of linearly independent and dependent subsets, Unique Expansion Theorem, Basis and the dimension of a vector space, properties and examples | Pages 27-31 of "A First Course in Linear Algebra" |
| 7 | Oct. 10 | Linear Operators and their examples, Thm: The image and inverse image of subspaces under a linear operator are subspaces, Null space, the dimension theorem. Linear equations and the existence of their solutions. | Pages 39-44 of "A First Course in Linear Algebra" |
| 8 | Oct. 12 | Uniqueness of solutions of general linear equations; Matrix representation of linear operators acting in finite-dimensional vector spaces; Algebra of linear operators | Pages 44-52 of "A First Course in Linear Algebra" |
| 9 | Oct. 17 | Application of the algebra of linear operators to linear second order ordinary differential equations (method of factorization for homogeneous equations with constant coefficients); Invertible operators, linearity of the inverse of a linear operator, onto-ness of invertible operators acting in a finite-dimensional vector space, isomorphisms and their role in classifying vector spaces | Pages 52-57 of "A First Course in Linear Algebra" |
| 10 | Oct. 19 | Addition and scalar multiplications of matrices, isomorphism between the vector space of linear operators defined between finite-dimensional vector spaces and the space of matrices, matrix multiplication, invertible matrices, inverse of invertible 2 x 2 matrices | Pages 63-69 of "A First Course in Linear Algebra" |
| Exam 1 | Oct. 21 | ||
| 11 | Oct. 24 | Determinants: Levi Civita symbol, Determinant of a square matrix and its basic properties, symmetric, Hermitian, upper-triangular, lower-triangular, and diagonal matrices, A simple method of calculating the determinant of 3x3 matrices, basic idea of the method of Gaussian elimination for calculating determinant of a square matrix | Pages 69-74 of "A First Course in Linear Algebra" |
| 12 | Oct. 26 | Calculating the inverse of an invertible matrix; Basis transformations: Transformation rule for the representation of the vectors and linear operators (similarity transformations) | Pages 74-76 of "A First Course in Linear Algebra" |
| Kurban Bayramý |
Nov.
07-11 |
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| 13 | Nov. 14 | Systems of linear algebraic equations, existence and uniqueness of solutions, Cramer's rule; | Pages 76-80 of "A First Course in Linear Algebra" |
| 14 | Nov. 16 | Eigenvalues and eigenvectors of a linear operator and a square matrix, the case of 2x2 matrices, an example with a single linearly-independent eigenvector, linear-independence of eigenvectors with different eigenvalues, diagonalizable operators and matrices, diagonalization procedure, example. | Pages 334-339 of Kreyszig |
| 15 | Nov. 21 | Inner product spaces: inner product, its associated norm and metric, examples; unit vectors and and orthogonal subsets, proof of linear-independence of orthogonal subsets, orthonormal bases | Pages 325-327 of Kreyszig |
| 16 | Nov. 23 | Extra PS | |
| Exam 2 | Nov. 25 | ||
| 17 | Nov. 28 | Representation of vectors and linear operators in an orthonormal basis of a finite-dimensional inner product space. Isometries and unitary operators. The proof of the existence of a unitary operator mapping every finite-dim. inner product space and the Euclidean space F^2. Convergence of sequences and series in an inner product space, dense subsets and approximation of functions, e.g., complex Fourier series |
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| 18 | Nov. 30 | Matrix representation of linear operators in an orthonormal basis; Symmetric operators and their matrix representation in orthonormal bases; Linear Ordinary Differenatial Eqs (ODEs): Solution and its existence and uniqueness theorem, solution of the first order equations, form of the general solution of an nth order linear ODE. | Pages 26-29 of Kreyszig |
| 19 | Dec. 05 | Second order linear ODEs: Wronskien and Abel's theorem, the form of the general solution, reduction of order, equations with constant coefficients. | Pages 45- 47, 50-56, 73-78 of Kreyszig |
| 20 | Dec. 07 | Solution of non-homogeneous first order linear ODEs and the notion of a Green's function, extension to non-homogeneous second order equations (method of variation of parameters) and the derivation of the formula for the Green's function | Pages 98-104 of Kreyszig |
| 21 | Dec. 12 | Power series and their convergence, Ratio test, (real) analytic functions. Power series solution of 2nd order linear ODE about an ordinary point, Hermite's eqn and Hermite polynomials | Pages 166-182 of Kreyszig |
| 22 | Dec. 14 | Frobenius method | Pages 182-188 of Kreyszig |
| 23 | Dec. 19 | Boundary-value problems: Dirichlet, Neumann, and Sturm-Liouville boundary conditions, Sturm-Liouville problem, orthogonal functions | Pages 203-209 of Kreyszig |
| 24 | Dec. 21 | Systems of first order linear ODE's: The existence and uniqueness theorem, Fundamental matrix, General form of the homogeneous and non-homogeneous systems and the matrix Green's function. | Pages 136-139 & 159-161 of Kreyszig |
| Exam 3 | Dec. 23 | ||
| 25 | Dec. 26 | Solution of homogeneous systems with constant coefficients: The case that the matrix of coefficients has real or complex simple eigenvalues. | Kreyszig: 139-146 |
| 26 | Dec. 28 | Solution of homogeneous systems with constant coefficients: The case that the matrix of coefficients has repeated eigenvalues; Solving a non-homogenous system of linear ODEs using the (matrix) Green's function. | Kreyszig: 159-165 |
Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.